The Mass of the Milky Way: Limits from a Newly Assembled Set of Halo Objects?

A&A 397, 899–911 (2003) Astronomy DOI: 10.1051/0004-6361:20021499 & c ESO 2003 Astrophysics

The mass of the Milky Way: Limits from a newly assembled set of halo objects?

T. Sakamoto1, M. Chiba2, and T. C. Beers3

1 Department of Astronomical Science, The Graduate University for Advanced Studies, Mitaka, Tokyo 181-8588, Japan 2 National Astronomical Observatory, Mitaka, Tokyo 181-8588, Japan 3 Department of Physics & Astronomy, Michigan State University, East Lansing, MI 48824, USA

Received 7 August 2002 / Accepted 14 October 2002

Abstract. We set new limits on the mass of the Milky Way, making use of the latest kinematic information for Galactic satellites and halo objects. Our sample consists of 11 satellite galaxies, 137 globular clusters, and 413 field horizontal-branch (FHB) stars up to distances of 10 kpc from the Sun. Roughly half of the objects in this sample have measured proper motions, permitting the use of their full space motions in our analysis. In order to bind these sample objects to the Galaxy, their rest-frame velocities must be lower than their escape velocities at their estimated distances. This constraint enables us to show that the mass estimate of the Galaxy is largely affected by several high-velocity objects (Leo I, Pal 3, Draco, and a few FHB stars), not by a single object alone (such as Leo I), as has often been the case in past analyses. We also find that a gravitational potential that gives rise to a declining rotation curve is insufficient to bind many of our sample objects to the Galaxy; a possible lower limit on the mass of the Galaxy is about 2.2 1012 M . To be more quantitative, we adopt a Bayesian likelihood approach to reproduce the observed distribution of the current× positions and motions of the sample, in a prescribed Galactic potential that yields a flat rotation curve. This method enables a search for the most likely total mass of the Galaxy, without undue influence in the final result arising from the presence or absence of Leo I, provided that both radial velocities and proper motions are used. Although the best mass estimate depends somewhat on the model assumptions, such as the unknown prior probabilities for the model parameters, the resultant systematic change in the mass estimate is confined to a relatively narrow range of a few times 11 +0.5 12 10 M , owing to our consideration of many FHB stars. The most likely total mass derived from this method is 2.5 1.0 10 M +0.4 12 − × (including Leo I), and 1.8 0.7 10 M (excluding Leo I). The derived mass estimate of the Galaxy within the distance to the − × +0.0 11 Large Magellanic Cloud ( 50 kpc) is essentially independent of the model parameters, yielding 5.5 0.2 10 M (including +0.1 11 ∼ − × Leo I) and 5.4 0.4 10 M (excluding Leo I). Implications for the origin of halo microlensing events (e.g., the possibility of brown dwarfs as− the× origin of the microlensing events toward the LMC, may be excluded by our lower mass limit) and prospects for more accurate estimates of the total mass are also discussed.

Key words. Galaxy: halo – Galaxy: fundamental parameters – Galaxy: kinematics and dynamics – stars: horizontal-branch

1. Introduction the extent over which such dark-matter-dominated mass distributions apply for most galaxies, including our own, is of great Over the past decades, various lines of evidence have revealed importance for understanding the role of dark matter in galaxy that the mass density in the Milky Way is largely dominated by formation and dynamical evolution. In particular, the mass es- unseen dark matter, from the solar neighborhood to the outer timate of the Galaxy is closely relevant to understanding the reaches of the halo (e.g., Fich & Tremaine 1991). Moreover, the origin of the microlensing events toward the Large Magellanic presence of a dark component similar to that found in our own Cloud (LMC) (e.g., Alcock et al. 2000; Alcock et al. 2001). Galaxy appears to be a generic feature in external galaxies, as inferred from, e.g., ﬂat rotation curves in their outer parts, the While mass estimates of external galaxies can, in princi- presence of (a gravitationally bound) hot plasma in early-type ple, be obtained in a straightforward fashion using various dy- galaxies, and the observed gravitational lensing of background namical probes, the total mass of the Galaxy remains rather sources (e.g., Binney & Tremaine 1987). A determination of uncertain, primarily due to the lack of accurate observational information for tracers located in its outer regions, where the Send oﬀprint requests to: T. Sakamoto, dark matter dominates. The precise shape of the outer rotation e-mail: [email protected] curve, as deduced from H II regions and/or H I gas clouds (e.g., ? Table 1 is only available at the CDS via anonymous ftp to Honma & Sofue 1997), is still uncertain because its determina- cdsarc.u.strasbg.fr (130.79.128.5)orvia tion requires knowledge of accurate distances to these tracers http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/397/899 (Fich & Tremaine 1991). Also, interstellar gas can be traced

Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20021499 900 T. Sakamoto et al.: The mass of the Milky Way onlyupto 20 kpc from the Galactic Center, and hence pro- In this paper we re-visit the mass determination of the vides no information∼ concerning the large amount of dark mat- Galaxy, based on a sample of 11 satellite galaxies, 137 globular ter beyond this distance. clusters, and 413 FHB stars, out of which 5 satellite galaxies, The most suitable tracers for determination of the mass dis- 41 globular clusters, and 211 FHB stars have measured proper tribution in the outer halo of the Galaxy are the distant lu- motions. First, we investigate the lowest possible mass that the minous objects, such as satellite galaxies, globular clusters, Galaxy may have, adopting the requirement that the rest-frame and halo stars on orbits that explore its farthest reaches (e.g., velocities of observed sample objects be less than their es- Miyamoto et al. 1980; Little & Tremaine 1987; Zaritsky et al. cape velocities at their present distance from the Galactic cen- 1989; Kochanek 1996; Wilkinson & Evans 1999, hereafter ter (e.g., Miyamoto et al. 1980; Carney et al. 1988). Secondly, WE99). However, the limited amount of data presently avail- the most likely mass of the Galaxy is calculated, based on a able on the full space motions of these tracers, and the small Bayesian likelihood analysis that seeks to reproduce both the size of the available samples, have stymied their use for an ac- current positions and velocities of the sample objects (e.g., curate determination of the Galaxy’s mass. In particular, most Little & Tremaine 1987; Kochanek 1996; WE99). Because our previous mass estimates (except for WE99, see below) depend sample of tracers is, by far, the largest and most accurate one quite sensitively on whether or not a distant satellite, Leo I, is presently available, it is possible to place more reliable limits bound to the Galaxy. Leo I has one of the largest radial ve- on the total mass of the Galaxy. In Sect. 2 we describe our locities of the known satellites, despite its being the second sample objects and the assembly of their kinematic data. In most distant satellite from the Galaxy (Mateo 1998; Held et al. Sect. 3 we discuss the influence of the adopted membership 2001). As a consequence, estimates of the total mass of the of our sample on the mass determination. In Sect. 4 we obtain Galaxy are much more uncertain (by as much as an order of the most likely total mass of the Galaxy based on a Bayesian magnitude) than, for instance, the value of the circular speed likelihood analysis. In Sect. 5 we discuss implications for the in the solar neighborhood (Kerr & Lynden-Bell 1986; Fich & origin of the halo microlensing events toward the LMC and the Tremaine 1991; Miyamoto & Zhu 1998; Méndez et al. 1999). mass estimate of the Local Group, and consider the prospects for more obtaining more accurate estimates of the total mass of Recently, by making use of both the observed radial veloc- the Galaxy in the near future. ities and proper motions of six distant objects, WE99 demonstrated that the use of full space motions can provide a reliable mass estimate of the Galaxy without being largely affected by 2. Data the presence or absence of Leo I. They also argued that the pri- We consider a sample of objects that serve as tracers of the mary uncertainties in their mass estimate arose from the small Galactic mass distribution consisting of 11 satellite galaxies, size of the data set and the measurement errors in the full space 137 globular clusters, and 413 FHB stars. In the case of the motions, especially the proper motions. This work motivated satellite galaxies, all of the basic information for our kinematic us to investigate a much larger data set, with more accurate analysis, i.e., positions, heliocentric distances, and heliocen- kinematic information, to set tighter limits on the mass of the tric radial velocities, are taken from the compilation of Mateo Galaxy. Specifically, as we show below, there are two objects (1998). For the globular clusters, we adopt the information pro- among the WE99 sample (Draco and Pal 3) that have relatively vided by Harris (1996), including their positions and heliocen- large velocity errors, yet still play crucial roles in a determina- tric radial velocities, their metal abundances, [Fe/H], and the tion of the Galaxy’s mass, so the addition of more (and better) apparent magnitude of the clusters’ horizontal branch (HB). data is important. The catalog of Wilhelm et al. (1999b) is our source of simi- Over the past few years, the number of distant satellite lar information for the FHB stars. We obtain an internally con- galaxies and globular clusters with available proper motions sistent set of distance estimates for the globular clusters and has gradually increased (e.g., Mateo 1998; Dinescu et al. 1999; the FHB stars from the recently derived relationship between Dinescu et al. 2000; Dinescu et al. 2001). In addition, an- the absolute magnitude of the HB, MV (HB), and [Fe/H], by other tracer population that is suitable for exploring mass es- Carretta et al. (2000), timates of the Galaxy has become available from the exten- MV (HB) = (0.18 0.09)([Fe/H] + 1.5) + (0.63 0.07). (1) sive compilation of A-type metal-poor stars by Wilhelm et al. ± ± (1999b), which provided radial velocity measurements, as well Clearly, we have assumed that there is no large offset be- as estimates of the physical parameters of these stars (e.g., tween the absolute magnitudes of FHB stars and their counter- [Fe/H], Teff,logg). Among the Wilhelm et al. sample, the part HB stars in the globular clusters (a view also supported luminous FHB stars are the most useful mass tracers, both by the recent work of Carretta et al. 2000). Figure 1 shows because of their intrinsic brightness, and the fact that accu- the spatial distribution of the globular clusters, satellite galax- rate distance determinations can be inferred from their absolute ies, and FHB stars on the plane perpendicular to the Galactic magnitudes on the horizontal branch (e.g., Carretta et al. 2000). disk, where X axis connects the Galactic center (X = 0) and Moreover, there exist proper-motion measurements for many of the Sun (X = 8.0 kpc). The filled and open symbols denote these stars, provided by both ground- and space-based proper- the objects with and without proper-motion measurements, re- motion catalogs (Klemola et al. 1994; Röser 1996; Platais et al. spectively. Satellite galaxies are the most distant tracers, with 1998; Hog et al. 2000), from which full space motions may be Galactocentric distances, r, greater than 50 kpc. The globular derived. clusters extend out to almost r = 40 kpc, while the present T. Sakamoto et al.: The mass of the Milky Way 901

(a) data for the LMC, Sculptor, and Ursa Minor are taken from WE99, whereas those for Sagittarius and Draco are taken from Irwin et al. (1996) and Scholz & Irwin (1994), respectively. 200 The proper motions for most of the globular clusters have been compiled by Dinescu et al. (1999). We adopt the data from this source, except for two globular clusters with recently revised proper-motion measurements (NGC 6254: Chen et al. 2000; 0 NGC 4147: Wang et al. 2000), and for three additional globular clusters compiled recently (Pal 13: Siegel et al. 2000; Pal 12: Dinescu et al. 2000; NGC 7006: Dinescu et al. 2001). Proper [kpc] motions for 211 of the FHB stars in the Wilhelm et al. (1999b) Z sample are available from one or more existing proper-motion -200 catalogs. These include the STARNET Catalog (R¨oser 1996), the Yale-San Juan Southern Proper Motion Catalog (SPM 2.0: Platais et al. 1998), the Lick Northern Proper Motion Catalog (NPM1: Klemola et al. 1994), and the TYCHO-2 Catalog (Hog -400 et al. 2000). Many of these FHB stars have been independently measured in two or more catalogs, so that by combining all -200 0 200 400 measurements one can reduce the statistical errors, as well as minimize any small remaining systematic errors in the individ- X [kpc] ual catalogs, as was argued in Martin & Morrison (1998) and Beers et al. (2000). (b) We estimate average proper motions, <µ>,andtheirer- 15 rors, <σµ >, weighted by the inverse variances, as

n n 2 2 10 <µ> = µi/σ / 1/σ , (2)  µi   µi  Xi=1 Xi=1      n  1/2   −   2 5 <σµ > = 1/σ , (3)  µi  Xi=1     0 where n denotes the number of measurements. Table 1 (avail- [kpc]

Z able at the CDS) lists these compilations, as well as the estimated distances to the FHB stars, where r and RV denote the -5 Galactocentric distances and heliocentric radial velocities, respectively. Typical errors in the reported proper-motion mea- 1 surements range from 1 to 5masyr− for individual field -10 stars, whereas those for the satellite∼ galaxies and globular clus- 1 1 ters are about 0.3 mas yr− and 1 mas yr− , respectively. We assume a circular rotation speed for the Galaxy of -15 1 -10 -5 0 5 10 15 20 VLSR = 220 km s− at the location of the Sun (i.e. R = 8.0 kpc along the disk plane) and a solar motion of (U, V, W )= ( 9, 12, X [kpc] 1 − 7) km s− (Mihalas & Binney 1981), where U is directed out- Fig. 1. Spatial distributions of satellite galaxies (squares), globular ward from the Galactic Center, V is positive in the direction of clusters (circles), and FHB stars (triangles) on the plane perpendic- Galactic rotation, and W is positive toward the North Galactic ular to the Galactic disk, where the X axis connects the Galactic cen- Pole1. We then calculate the space motions of our tracers, as ter (X = 0) and the Sun (X = 8.0 kpc). The filled and open sym- well as the errors on these motions, fully taking into account bols denote the objects with and without available proper motions, the reported measurement errors in the radial velocities of the respectively. The plus sign in panel b) denotes the position of the Sun, 1 individual satellite galaxies (typically a few km s− ), adopting (X, Y) = (8.0, 0). 1 a typical radial-velocity error for other objects (10 km s− ), the measurement errors assigned to the proper motions of each object (when available, adopting a mean error for the sample of FHB stars are confined to locations within 10 kpc source catalog when not), and distance errors for the satellite of the Sun. Thus, our sample objects are widely, though not uniformly, distributed throughout the volume of the Galaxy. 1 Dehnen & Binney (1998) derived the solar motion of (U, V, W) = 1 Among these sample objects there exist proper-motion ( 10.0, 5.3, 7.2) km s− based on Hipparcos data. Because the differ- − measurements for 5 of the satellite galaxies, 41 of the glob- ence between this and the currently adopted solar motion is only a 1 ular clusters, and 211 of the FHB stars. The proper motion few km s− , it gives little influence on the mass estimate. 902 T. Sakamoto et al.: The mass of the Milky Way galaxies (10% relative to the measured ones), or as obtained 300 from Eq. (1) for the globular clusters and FHB stars. It is worth noting that the reported proper motions of the 250 FHB stars in our sample may yet contain unknown systemat- 200 ics with respect to their absolute motions in a proper reference frame; this caution applies to the globular clusters and satel- 150 Model A, a =200 kpc lite galaxies as well. It is an important goal to make efforts to [km/s] c reduce the systematic, as well as random, errors in the proper V 100 Model A, a =20 kpc motions upon which studies of Galactic structure and kinematic Model B, r cut =170 kpc studies are based, using much higher precision astrometric ob- 50 servations than have been obtained to date. 0 0 5 10 15 20 3. Roles of the sample in the mass limits R [kpc]

If we model the Galaxy as an isolated, stationary mass distribu- Fig. 2. Rotation curves for Model A and Model B, parameterizations tion, and assume that all of our tracer objects are gravitationally of the mass distributions considered in this paper. See the text for more bound to it, then the rest-frame velocities of all objects, VRF, information on the nature of these models. must be less than their escape velocities, Vesc = 2ψ,whereψ denotes the gravitational potential of the Galaxy.p A number of (see their Table 3). In order to obtain a finite total mass we previous researchers have adopted this method for estimation assume the following modified logarithmic potential (corre- of the mass of the Galaxy (e.g., Fricke 1949; Miyamoto et al. sponding to an isothermal-like density distribution) for the dark 1980; Carney et al. 1988; Leonard & Tremaine 1990; Dauphole halo component: & Colin 1995). Note that the mass estimate obtained in this 2 2 fashion is largely dominated by only a small number of high- v0 log[1 + (r/d) ] ψ0, at r < rcut ψhalo(r) = 2 rcut c − (5) velocity objects, hence the mass that is derived depends rather ( 2v , at r rcut, − 0 r 1+c ≥ sensitively on the selection criteria adopted for such objects. In the present section, instead of deriving an exact mass estimate, 2v2 3 + r/d ρ(r) = 0 , (6) we follow this procedure for the purpose of elucidating the role 4πGd2 (1 + r/d)3 of sample selection in setting limits on the mass of the Galaxy. To this end we consider two different mass models, in order where ψ0 is defined as to investigate the difference in the mass limits obtained by the 2 2 ψ0 = v [log(1 + c) + 2c/(1 + c)], c = (rcut/d) , (7) use of different potentials. Our models, hereafter referred to as 0 1 Model A and B, are the same as those adopted in WE99 and and we adopt v0 = 128 km s− and d = 12 kpc (Dinescu Johnston et al. (1995) (and also used by Dinescu et al. 1999), et al. 1999). This model contains one free parameter, namely respectively. the cutoff radius of the dark halo, rcut. Figure 2 shows the ro- Model A has spherical symmetry, and results in a flat rota- tation curves for 0 R 20 kpc, provided by Model A with ≤ ≤ tion curve in the inner regions of the Galaxy. The gravitational a = 200 kpc (thick solid line) and Model B with rcut = 170 kpc potential and mass density are given as: (thin solid line), where both curves shown at R 20 kpc re- ≤ main unchanged as long as a, rcut 20 kpc. The circular speed GM √r2 + a2 + a R R 1 ψ(r) = log , at = is 220 km s− for both mass models. Also shown is a  r  the declining rotation curve with increasing radius, as obtained     a V = 1 M a2  from Model A with = 20 kpc and LSR 211 km s− (dashed ρ(r) = , (4) line). 4π r2(r2 + a2)3/2 We plot, in Figs. 3a and 3b, the relationship between the de- where a is the scale length of the mass distribution, and M is rived escape velocities, Vesc, and the rest-frame velocities, VRF, the total mass of the system. The central density of this model when we adopt Model A with a = 195 kpc and Model B with 2 5 is cusped (like r− ) and falls off as r− for r a.UsingEq.(4), rcut = 295 kpc, respectively. For the objects without available 2 2 2 1/2 the circular rotation speed is given as Vc = GM/(r + a ) ,so proper motions (open symbols), we adopt the radial velocities 1 by setting Vc at r = R as VLSR = 220 km s− in our standard alone as measures of VRF, hence their estimated space veloci- case, it follows that this model contains one free parameter, a, ties are only lower limits. The solid line denotes the boundary to obtain M. between the objects that are bound (below the line) and un- Model B consists of more realistic axisymmetric potentials bound (above the line) to the Galaxy, respectively, where the with three components (the bulge, disk, and dark halo) that positions of the data points relative to the solid line depend on reproduce the shape of the Galactic rotation curve (Johnston the choice of a or rcut. As these figures clearly indicate, the se- et al. 1995). The bulge and disk components we adopt are lection of the smallest value of a or rcut that places the sample those represented by Hernquist (1990) and Miyamoto & Nagai inside the bound region, or equivalently the lower limit on the (1975) potentials, respectively. All of the parameters included mass of the Galaxy, is controlled by a few high-velocity ob- in these potentials are taken from Dinescu et al. (1999) jects located near the boundary line at each respective radius T. Sakamoto et al.: The mass of the Milky Way 903

(a) 800 by Draco and Pal 3 is basically the same as that provided by Leo I. This may explain the result of WE99, which showed 700 that the mass determination is made insensitive to Leo I if the proper motion data of satellite galaxies and globular clusters 600 Draco are taken into account. However, as Fig. 3 indicates, the velocity errors for Draco and Pal 3 are quite large, so these ob- 500 jects place only weak constraints on the mass estimate. (3) If we consider the proper motions of FHB stars, then some FHB 400 stars having high velocities provide the basically the same con- Pal 3 straint on the Galaxy’s mass as do Leo I, Draco, and Pal 3. RF [km/s] V 300 These properties suggest that the inclusion of FHB stars with available proper motions is crucial, and that they provide con- 200 straints on the mass limit of the Galaxy that depend on neither Leo I the inclusion or absence of Leo I nor on the large velocity er- 100 rors for Draco and Pal 3. As mentioned above, a determination of the lower mass 0 limit for the Galaxy, using escape velocities, inevitably depends 0 100 200 300 400 500 600 700 800 on the selection of a few apparently high-velocity objects from Vesc (r )[km/s] a much larger sample of tracers. While a mass estimate independent of this selection effect will be obtained in Sect. 4, we (b) 800 seek first to obtain a rough measure for the lower mass limit, i.e., the smallest a or rcut that encloses Pal 3, Draco, and the 700 four highest-velocity FHB stars (for which we mark asterisks after their names in Table 1) inside the bound region, based on 600 a weighted least-squares fitting procedure (weights being in- Draco versely proportional to the velocity errors). This exercise yields a +160 r +335 500 = 195 85 kpc for Model A, and cut = 295 145 kpc for Model B.− Using these values, the lower limits to the− total mass, +1.8 12 400 M, of the Galaxy may be given as 2.2 1.0 10 M for Model A Pal 3 and 2.2+2.6 1012 M for Model B, respectively.− × Thus, the dif- [km/s]

RF 1.1 − × V 300 ference in the derived mass limits is not significant, as long as the rotation curve at outer radii is approximately constant at 1 200 the adopted value of 220 km s− . It also suggests that the flat- tened nature of the Model B potential, due to the presence of Leo I 100 the disk component, does not affect the results significantly – the high-velocity tracers are located at large Galactocentric dis- 0 tances and/or their orbits largely deviate from the disk plane. 0 100 200 300 400 500 600 700 800 In addition to the above experiments, we also consider a mass model that yields a declining rotation curve at outer Vesc (R,Z )[km/s] radii, as was proposed by Honma & Sofue (1997) from their Fig. 3. a) The relation between escape velocities, Vesc, and space ve- H I observations. We adopt Model A with a = 20 kpc and locities, V , for Model A with a = 195 kpc and V = 220 km s 1. 1 RF LSR − VLSR = 211 km s− at R = R , so as to yield the declining The symbols are the same as those in Fig. 1. The solid line denotes the rotation curve at R > R (dashed line in Fig. 2), which is remi- boundary between the gravitationally bound and unbound objects – niscent of the result in Honma & Sofue (1997). Figure 4 shows those in the region below the line are bound to the Galaxy. For the sake the V vs. V relationship that follows from adoption of this of clarity, velocity errors are plotted for only the high-velocity objects RF esc relevant to the mass estimate. b) Same as panel a) but for Model B model. As is evident, the total mass obtained from a model that leads to a declining rotation curve is insufficient to bind many with rcut = 295 kpc. of our sample objects to the Galaxy.

(or corresponding ψ). These objects include Leo I (for which only radial velocity information is available), Draco, Pal 3, and 4. Mass determination based on a Bayesian several high-velocity FHB stars. This highlights the following likelihood method important properties of the derived mass limits: (1) If the proper 4.1. Method motions of all objects are unavailable, then the mass estimate sensitively depends on the presence or absence of Leo I, as has To obtain a more quantitative measure of the mass of the been noted in previous studies. (2) Compared to case (1), if Galaxy, we examine an alternative method that takes into ac- the available proper motions of the satellite galaxies and glob- count all of the positional and kinematic information of the ular clusters are taken into account, the constraint provided sample objects, in contrast to the use of the high-velocity 904 T. Sakamoto et al.: The mass of the Milky Way 800 160

700 140 120 600 data(all globular clusters Draco 100 ) r and satellite galaxies) 80 500 (< γ =3.3 N 60 400 a s =10 kpc 40

[km/s] Pal 3 RF V 300 20 0 200 Leo I 0 100 200 300 400 500 r [kpc] 100 Fig. 5. Cumulative number distribution, N(

Fig. 4. The relation between escape velocities, Vesc, and space veloc- function yields equal velocity dispersions in the orthogonal an- 1 ities, VRF, for Model A with a = 20 kpc and VLSR = 211 km s− .In gular directions, = , and a constant anisotropy this case, the rotation curve declines with increasing radii, as shown θ φ β = 1 / everywhere in the Galaxy. Our choice in Fig. 2 (dashed line). Note that, if this situation were to apply, many − θ r of the sample objects would be unbound to the Galaxy. of m = 2 in Eq. (9) (to be in accord with the WE99 work) limits the allowed range for the velocity anisotropy to 1.5 β 1 when proper motion data are considered, while the− use≤ of radial≤ tracers alone, as in the previous section. In this approach, a velocities alone sets no limit for tangential anisotropy [ , 1]. −∞ phase-space distribution function of tracers, F, is prescribed For ρs, we consider WE99’s two models: (a) Shadow for a specifically chosen ψ, and the model parameters included tracers following the mass density distribution obtained from in F and ψ are derived so as to reproduce the presently observed Model A (Eq. (4)), and (b) a power-law distribution as a func- positions and velocities of the tracers in the (statistically) most tion of r. The shadow-tracer model is given as: significant manner. The optimal deduced parameters relevant a2 to ψ then allow us to estimate the total mass of the Galaxy. This ρ (r) s , (10) s 2 2 2 3/2 method was originally proposed by Little & Tremaine (1987), ∝ r (r + as ) and further developed by Kochanek (1996) and WE99. where as is the scale length. The power-law model with index γ Based on the results presented in the previous section, we is given as: take Model A with spherical symmetry as the mass distribution of the Galaxy, which is sufficient for the following analysis. 1 ρs(r) (11) For the sake of simplicity, and also for ease of comparison with ∝ rγ · the previous studies by Kochanek (1996) and WE99, the phase- It should be noted that, since shadow tracers may be truncated space distribution function is taken to have the same anisotropic at the distance below the scale length of the mass distribution, form as that adopted in these studies. That is, it depends on the the scale length of the tracers, as, is generally different from the binding energy per unit mass, ε ( ψ v2/2), and the angular ≡ − scale length of the Galaxy’s mass, a. momentum per unit mass, l, in the following way: Using the 27 objects (satellite galaxies and globular clusters) at r > 20 kpc, WE99 derived a = 100 kpc and γ = 3.4 F(ε, l) = l 2β f (ε), (8) s − as the best fitting parameters for their spatial distribution. We where re-examine as and γ using our sample of all satellite galaxies and globular clusters. Note that the FHB stars are ex- β 3/2 2 − d cluded in this determination of a and γ, as they have not (yet) f (ε) = s π3/2Γ[m 1/2 + β]Γ[1 β] dε been completely surveyed over the Galactic volume. We obtain − − ε m 2β as = 10 kpc and γ = 3.3 as the best-fit values, based on a sim- d r ρs β 3/2+m dψ (ε ψ) − , (9) ple K–S test of the observed vs. predicted distribution functions × Z dψm − 0 (see Fig. 5). If we exclude the globular clusters at r 10 kpc, ≤ where ρs is the tracer density distribution, Γ is the gamma func- for which the spherical symmetry assumption may be question- tion, and m is an integer whose value is chosen such that the able due to the presence of the disk globular clusters, we obtain integral in Eq. (9) converges (e.g., Dejonghe 1986; Kochanek as = 50 kpc and γ = 3.4. Thus, as depends sensitively on the 1996). In the spherical model, this form of the distribution adopted range of radius (i.e., on the selection of the sample), T. Sakamoto et al.: The mass of the Milky Way 905 whereas γ basically remains unchanged. Therefore, we focus moment the available proper-motion information. Specifically, our attention on the results obtained from the power-law rep- we focus on the difference in the mass estimate arising from resentation for the tracer population; the shadow-tracer pop- the presence or absence of Leo I. Figure 6 shows the likelihood ulation is also examined for the purpose of comparison with contours in the mass-anisotropy (M β) plane for the case of WE99. To see the dependence of the mass estimate on these pa- a power-law tracer population with −γ = 3.4, where β is lim- rameters, we obtain estimates for two values of γ (3.4 and 4.0) ited to the range of 1.5 β 1. The solid and dashed lines − ≤ ≤ and as (100 kpc and the scale length of the mass distribution, denote the presence and absence of Leo I, respectively. As is a), respectively. We note that the FHB stars are also expected to evident, the mass estimate sensitively depends on whether or follow a power-law form with γ 3.4, as inferred from other not Leo I is bound to the Galaxy, as has been noted in previ- halo field stars (e.g., Preston et al.' 1991; Chiba & Beers 2001). ous studies. Inclusion of Leo I yields a likely total mass that We calculate the likelihood of a particular set of model pa- is an order of magnitude greater than the case without Leo I. rameters (the scale length of the mass distribution, a,andthe Over the range of β we consider, the most likely value of M, 11 anisotropy parameter, β) given the positions, ri, and radial ve- with Leo I included, is 21.0 10 M , corresponding to a × locities, vri, or space velocities, vi, using Bayes’ theorem. The scale length a = 185 kpc, whereas excluding Leo I yields probability that the model parameters take the values a and β, M = 9.6 1011 M ,anda = 85 kpc. We note that the role × given the data (ri,v(r)i) and prior information I,is: of Leo I in the Galaxy’s mass estimate is also understandable from the escape-velocity argument; if only the sample radial 1 N P(a,βri,vr i, I) = P(a)P(β) P(ri,vr i a,β), (12) velocities are taken into account, Leo I alone determines the | ( ) N ( ) | Yi=1 best-fit boundary line VRF = Vesc in the VRF vs. Vesc diagram where N is the normalization factor (Kochanek 1996; WE99). (Fig. 3). The probabilities P(a)andP(β) denote the prior probability As is seen in Fig. 6, the high-probability region is biased distributions in a and β, respectively. Here, P(ri,v(r)i a,β) corre- toward the line β = 1.5. This bias arises from the specific | − sponds to the probability of finding an object at position ri mov- form of the phase-space distribution function, F(ε, l), given in ing with radial velocity v(r)i, or space velocity vi, for a particular Eq. (8), where the probability P(a,βri,v(r)i, I)ishighatlargeF. set of model parameters a and β. The complete expressions for We plot F in Fig. 7 for a set of r and |β (solid and dotted lines for P(ri,v(r)i a,β) are shown in Table 1 of WE99. To calculate this β = 1 and 1, respectively). It follows that F at high ε is larger probability| for the objects with full space velocities, we take for smaller− β, whereas F at low ε is larger for larger β.The into account their large errors relative to radial velocities alone range of ε corresponding to these two different cases depends (due to the observed proper-motion and assumed distance er- on r, as can be deduced from the comparison between panels a rors), by multiplying by an error convolution function of the and b in Fig. 7. Since our sample objects are mainly distributed form: in the region of higher ε (solid histograms for the sample with radial velocities), the probability is highest at smallest β. P(ri,vi a,β) = dvαdvδE1(vα)E1(vδ) | ZZ Following the above experiments, we drop the lower bound P(ri,vi,obs(vα,vδ) a,β), (13) of 1.5forβ, and search for the maximum probability at × | smaller− β. No maximum is found up to β = 20, although where (vα,vδ) are the tangential velocities along the right as- the large discrepancy in M between the cases with− and with- cension and declination coordinates, respectively, and E1 is the Lorentzian error convolution function, defined as: out Leo I remains. When we confine ourselves to the sample at r > β . 2 10 kpc, there exists a maximum probability at = 2 75 1 2σ 11 − E (v) = 1 , (14) (with Leo I), with a corresponding mass 32.0 10 M .For 1 2 2 11× √2πσ1 2σ1 + (v vobs) the sample at r > 20 kpc, we obtain 11.4 10 M at β = 0.8. − This clearly suggests that the best-fitting×β, obtained from the where σ1 is defined as σ1 = 0.477σ for the calibrated error estimate σ (see WE99). analysis when only radial velocities are considered, is rather The prior probability in the velocity anisotropy, β, is taken sensitive to the range of r employed in the sample selection. to be of the form P(β) 1/(3 2β)n,wheren = 0and2 This in turn affects the number distribution, N(ε), which is rel- correspond to a uniform∝ prior and− a uniform energy prior, re- evant to the likely range of F (Fig. 7). spectively (Kochanek 1996; WE99). Larger values of n yield a With these unavoidable limitations of the present sample in larger weight towards radial anisotropy. For the prior probabil- mind, Table 2 summarizes the likelihood results for the limited ity in a, P(a), we adopt 1/a and 1/a2 (WE99). range of 1.5 β 1, obtained for power-law and shadow Using the routine AMOEBA in Numerical Recipes (Press tracers, using− a≤ variety≤ of different priors on a and β.Themost et al. 1992), we search for a set of model parameters, a and β, likely value of β is 1.5 for all cases, for the reasons described − that maximize the probability P(a,βri,vr i, I). The total mass above. We note that the current mass estimate is rather insensi- | ( ) of the Galaxy, M, is then derived from the parameter a. tive to the β prior. As the β prior decreases, the estimated mass generally increases, and the best-fitting β decreases, because the small β prior is biased toward more tangentially anisotropic 4.2. Results velocity distributions than is the large β prior. However, since Initially, we apply the Bayesian likelihood method, making use most of our sample have high ε, the best-fitting β remains of only the radial velocities of the objects, setting aside for the 1.5 regardless of whether we adopt the uniform prior or − 906 T. Sakamoto et al.: The mass of the Milky Way

Table 2. Likelihood results for only the radial velocities.

a a a

ÓÖ a a ÔÖiÓÖ ¬ ÔÖiÓÖ ÄeÓ Á b e×Ø ¬ b e×Ø a ´kÔ cµ b e×Ø Å Å ´< 5¼kÔ cµ Å ´< ½¼¼kÔ cµ

ÈÓÛeÖ¹ÐaÛÌÖaceÖ×

=¿:4 ½=a EÒeÖgÝ Ye× ½:5 ½85 ¾½º¼ 5º4 9º9

ÆÓ ½:5 85 9º6 4º9 7º¿

=¿:4 ½=a EÒeÖgÝ Ye× ½:5 ¾½¼ ¾4º¼ 5º5 ½¼º¼

ÆÓ ½:5 95 ½½º¼ 5º¼ 7º8

=¿:4 ½=a ÍÒifÓÖÑ Ye× ½:5 ½85 ¾½º¼ 5º4 9º9

ÆÓ ½:5 85 9º6 4º9 7º¿

=4:¼ ½=a EÒeÖgÝ Ye× ½:5 ½95 ¾¾º¼ 5º5 ½¼º¼

ÆÓ ½:5 9¼ ½¼º¼ 4º9 7º6

ËhadÓÛÌÖaceÖ×

a = ½¼¼ ½=a EÒeÖgÝ Ye× ½:5 ¾½¼ ¾4º¼ 5º5 ½¼º¼

ÆÓ ½:5 85 9º6 4º9 7º¿

a = ½¼¼ ½=a EÒeÖgÝ Ye× ½:5 ¾4¼ ¾7º¼ 5º5 ½¼º¼

ÆÓ ½:5 95 ½½º¼ 5º¼ 7º8

a = ½¼¼ ½=a ÍÒifÓÖÑ Ye× ½:5 ¾½¼ ¾4º¼ 5º5 ½¼º¼

ÆÓ ½:5 85 9º6 4º9 7º¿

a = a ½=a EÒeÖgÝ Ye× ½:5 ¾¼5 ¾¿º¼ 5º5 ½¼º¼

ÆÓ ½:5 9¼ ½¼º¼ 4º9 7º6

a ½½

AÐÐ Ña××e× aÖe iÒ ÙÒiØ× Óf ½¼ Å º ¬ the uniform-energy prior for β. This property makes the mass index γ, unknown prior probabilities for a and β,aswellason estimate insensitive to the β prior. the range of r used in the sample selection, resulting in small Now we apply the Bayesian likelihood method to the sub- changes in the mass estimates over a range of only a few times sample of objects with both radial velocities and proper mo- 1011 M . tions available, and consider the derived space motions. In con- To estimate the typical errors in this mass determina- trast to the above case, where we used radial velocities alone, tion that are associated with the measurement errors of the we find that the maximum probability within the range of β 561 tracers we have analyzed, we have conducted Monte Carlo we consider is now bounded (Fig. 8a). This may be caused simulations, adopting the assumptions that typical errors in the 1 by the characteristic distribution of ε for the sample with full distances and radial velocities are 10%, and 10 km s− ,re- 1 space motions, as shown in Fig. 7 (dotted histogram). This fig- spectively, and that the proper-motion errors are 1 mas yr− 1 ure shows that there exists a larger fraction of low–ε stars than for globular clusters, 0.3 mas yr− for satellite galaxies, and 1 are found in the sample with radial velocities alone (solid his- 5masyr− for the FHB stars. We generated 561 data points tograms), so a larger β is preferred to achieve a larger F.The (including Leo I) drawn from Gaussian distribution functions mass estimate in this case is quite insensitive to the presence or centered on the observational data, and with dispersions set to absence of Leo I. Figure 8b shows the probabilities, as a func- the above typical errors. Given a true mass M, or scale length tion of M, with a fixed value of β = 1.25, for the case of a a (where we use M = 2.3 1012 M with a = 200 kpc), and power-law tracer population with γ =−3.4. Solid and dashed prior probabilities for a and× β (1/a2 and the uniform-energy lines denote the probabilities with and without Leo I, respec- prior, respectively), we calculate the most likely mass, M0,and tively. As is evident, the agreement between both probabilities compare it with an input true mass. Figure 9 shows the distribu- is significantly improved compared to the case when the ra- tion of the discrepancy between M0 and M, 100 (M0 M)/M, dial velocities are considered alone (Fig. 6b). When Leo I is obtained from 1000 realizations. The error distribution× − in the included, the most likely value of the total mass, M,andthe current mass estimate has a mean value shifted downward by scale length, a,are25.0 1011 M and 225 kpc, respectively. 20%, and a dispersion of half-width 20%. These values sug- Excluding Leo I yields M×= 18.0 1011 M and a = 160 kpc. gest that one might adopt an estimate of the systematic error Table 3 summarizes the various× results obtained when the on the order of 20%, and a random error of 20%. Exclusion proper motions of the objects are considered. This table illus- of Leo I does not influence the magnitude± of these errors. trates that, for all cases, the mass of the Galaxy obtained when It is worth noting that WE99 obtained roughly 100% sys- including Leo I is in good agreement with that obtained with- tematic errors, and 90% random errors in their∼ mass esti- out Leo I. Also, the mass estimate depends only weakly on the mate, which was based∼ on about 30 data points. The significant T. Sakamoto et al.: The mass of the Milky Way 907

Table 3. Likelihood results for the full space velocities.

a a a

ÓÖ a a ÔÖiÓÖ ¬ ÔÖiÓÖ ÄeÓ Á b e×Ø ¬ b e×Ø a ´kÔ cµ b e×Ø Å Å ´< 5¼kÔ cµ Å ´< ½¼¼kÔ cµ

ÈÓÛeÖ¹ÐaÛÌÖaceÖ×

=¿:4 ½=a EÒeÖgÝ Ye× ½:¾5 ¾¾5 ¾5º¼ 5º5 ½¼º¼

ÆÓ ½:¾5 ½6¼ ½8º¼ 5º4 9º6

=¿:4 ½=a EÒeÖgÝ Ye× ½:¾5 ¾55 ¾9º¼ 5º5 ½¼º¼

ÆÓ ½:¾5 ½75 ¾¼º¼ 5º4 9º8

=¿:4 ½=a ÍÒifÓÖÑ Ye× ½:¾5 ¾¾5 ¾5º¼ 5º5 ½¼º¼

ÆÓ ½:¾5 ½6¼ ½8º¼ 5º4 9º6

=4:¼ ½=a ÍÒifÓÖÑ Ye× ½:¿5 ¾45 ¾8º¼ 5º5 ½¼º¼

ÆÓ ½:¿5 ½75 ¾¼º¼ 5º4 9º8

ËhadÓÛÌÖaceÖ×

a = ½¼¼ ½=a EÒeÖgÝ Ye× ½:¾5 ¾9¼ ¿¿º¼ 5º5 ½½º¼

ÆÓ ½:¾5 ¾¼¼ ¾½º¼ 5º5 ½¼º¼

a = ½¼¼ ½=a EÒeÖgÝ Ye× ½:¾5 ¿5¼ ¿7º¼ 5º6 ½½º¼

ÆÓ ½:¾5 ¾4¼ ¾5º¼ 5º5 ½¼º¼

a = ½¼¼ ½=a ÍÒifÓÖÑ Ye× ½:¾5 ¾9¼ ¿¿º¼ 5º5 ½½º¼

ÆÓ ½:¾5 ¾¼¼ ¾½º¼ 5º5 ½¼º¼

a = a ½=a EÒeÖgÝ Ye× ½:¿ ¿½5 ¿5º¼ 5º6 ½½º¼

ÆÓ ½:¿ ¾¾¼ ¾5º¼ 5º5 ½¼º¼

a ½½

AÐÐ Ña××e× aÖe iÒ ÙÒiØ× Óf ½¼ Å º ¬ improvement of our mass estimate is mainly due to our consid- from this approach depends somewhat on model assumptions eration of a much larger data set that includes several hundred (prior probabilities for a and β and possibly the shape of F, FHB stars. see below), the resultant systematic change of the total mass As shown in Table 3, the most likely estimated total mass is confined to within a few times 1011 M . The most likely to- depends on model assumptions at a level of a few times tal mass of the Galaxy we derive is 2.5+0.5 1012 M .This 1.0 × 1011 M . When the model is fixed, the current large data set is in good agreement with the total mass− obtained by WE99 allows us to limit both systematic and random errors to a level (1.9+3.6 1012 M ) and that obtained from other methods (e.g., 1.7 × of about 20%. If we follow WE99’s procedure for the adoption Peebles− 1995, 2 1012 M ). Since the size of our tracer sample × of the most likely total mass, i.e., if we adopt the mass estimate is significantly larger than used in previous studies, both sys- that provides the smallest difference between the masses ob- tematic and random errors are reduced to a great extent. We +0.5 12 tained with Leo I and without Leo I, we obtain 2.5 1.0 10 M note that consideration of the numerous FHB stars plays a vital (Leo I included) and 1.8+0.4 1012 M (Leo I excluded).− × On role in this mass estimate, as demonstrated in Sect. 3. 0.7 × the other hand, the mass− estimate within the distance of the It is also worth noting that, if we fix the mass of the LMC (50 kpc) is quite robust, covering the narrow range 5.4 Galaxy equal to our most likely mass estimate, there is insuf- to 5.5 1011 M . ficient matter present to gravitationally bind the LMC, if we × adopt the recent proper-motion measurement by Anguita et al. (2000). These authors reported rather high proper motions, 5. Discussion and concluding remarks (µα cos δ, µδ) = (+1.7 0.2, +2.9 0.2), compared to previous ± ± We have placed new limits on the mass of the Galaxy, based on measurements, (µα cos δ, µδ) = (+1.94 0.29, 0.14 0.36) ± − ± a newly assembled set of halo objects with the latest available (Kroupa & Bastian 1997). Thus their results need confirmation proper-motion data. First, the comparison of their space veloc- from other studies2. ities with the escape velocities at their estimated distances al- The current work also implies that the Galactic rotation lowed us to show that the mass limits we obtained depend on curve at outer radii, R > R , does not decline out to at least neither the presence or absence of Leo I, nor on the large ve- R 20 kpc (as long as local disturbances to circular motions, locity errors for Draco and Pal 3; a possible a lower limit on the such∼ as warping motions and/or non-axisymmetric motions, are total mass of the Galaxy is about 2.2 1012 M . Secondly, a ignored). As illustrated in Fig. 2, a declining rotation curve Bayesian likelihood approach has∼ been× used to derive a total mass estimate for the Galaxy that is insensitive to the pres- 2 Since we finished our analysis here, a new study by Pedreros et al. ence or absence of Leo I, at least when proper motions are (2002) has been published (using the same method as Anguita el al. taken into account. Although the best mass estimate obtained 2000), that reaches the same conclusion as Kroupa & Bastian (1997). 908 T. Sakamoto et al.: The mass of the Milky Way

(a) (a)

100 5 80 r =10 kpc, β=-1 including Leo I 0 70 excluding Leo I r =10 kpc, β= 1 60 -5 ) ) l

M , 50

ε 11 N

( -10

10 ( F

40 ε ) (X10

10 r <20 kpc,v -15 r M

log 30 -20 20 r <20 kpc,v -25 1 RF 10 -1.5 -1.4 -1.3 -30 0 β 0 1 2 3 4 5 (b) ε (b) β =-1.5 5 10 1 including Leo I r =50 kpc, β=-1 excluding Leo I 0 r =50 kpc, β= 1 8 -5 ) l

0.5 , ε 6

-10 N ( Probability

F ( ε ) 10 -15 20

1 corresponding to a = 20 kpc and VLSR = 211 km s− fails to bind many sample objects to the Galaxy. The smallest possible proper motions of astronomical maser sources that are widely value for a to bind all objects in the isothermal-like density dis- distributed in the Galactic disk (Sasao 1996; Honma et al. tribution (Eq. (4)) is a = 195 kpc, yielding V 220 km s 1. 2000). VERA will reach unprecedented astrometric precision, LSR ' − In a more general context, the detailed shape of the rota- 10 µas, and will yield precise determinations of the Galactic ∼ tion curve at and beyond R = R reﬂects the interplay between constants R and VLSR. We note that whatever results are de- the disk and halo mass distributions, as this region is located rived for the rotation curve, the total mass of the Galaxy ought near the boundary of both components. Thus, determining the to be larger than 1012 M , in order to bind the more distant rotation curve at R < R < 15 kpc will set useful limits on stellar objects. the mass distribution ∼ in the∼ inner parts of the Galaxy. Indeed, Our estimate for the mass of the Galaxy inside 50 kpc, i.e., +0.0 11 the Japanese project VLBI Exploration of Radio Astrometry within the distance of the LMC, is 5.5 0.3 10 M (Leo I +0.1 11 − × (VERA) will be able to determine both inner and outer rotation included) and 5.3 0.4 10 M (Leo I excluded). The er- curves from the measurement of trigonometric parallaxes and ror estimates are calculated− × from the maximum and minimum T. Sakamoto et al.: The mass of the Milky Way 909 (a) 500

100 ∆r =10% ∆ 400 Vr =10 km/s ∆µ g = 1 mas/yr ∆µ =5 mas/yr F

) 300 ∆µ s =0.3 mas/yr M

11 10 N

(X10 including Leo I 200

M excluding Leo I 100

1 -1.5 -1.4 -1.3 -1.2 -1.1 -1 0 β -100 -50 0 50 100 relative error (%) (b) Fig. 9. An approximate error distribution of the mass estimate caused by the typical measurement errors of the data. The abscissa denotes β=-1.25 the relative error in mass, 100 (M0 M)/M,whereM0 is the mass 1 calculated by a Monte Carlo method× and− M is the input true value. See including Leo I text for more details. excluding Leo I

Galactic disk (Alcock et al. 2001). More direct observations for 0.5 identifying lensing objects are required to settle this issue.

Probability Once the total mass of the Galaxy is fixed, it is possible to place a useful constraint on the mass of the Local Group. Most of the mass in the Local Group is concentrated in M 31 and the Galaxy. The total mass of M 31 can be estimated 0 from the positions and radial velocities of its satellite galaxies, 1 10 100 globular clusters, and planetary nebulae (Evans & Wilkinson 11 M (X10 M ) 2000; Côté et al. 2000; Evans et al. 2000). If we take it to be +1.8 12 1.2 0.6 10 M (Evans & Wilkinson 2000), the mass of the Fig. 8. a) Likelihood contours in the plane of the mass, M,andveloc- Local− Group× is 3.7 1012 M . This is in good agreement with ity anisotropy, β, obtained from an analysis that uses both radial ve- the estimate by∼ Schmoldt× & Saha (1998), (4–8) 1012 M , locities and proper motions. Solid and dashed curves show the results based on modified variational principles. × including Leo I and excluding Leo I, respectively; the cross and the To set tighter limits on the total mass of the Galaxy we re- asterisk show the maxima of the probabilities for each case. Contours quire more accurate proper-motion measurements for a greater are plotted at heights of 0.32, 0.1, 0.045, and 0.01 of the peak height. The spatial distribution of a tracer population is assumed to follow a number of objects at large Galactocentric distances. The high- power-law form with γ = 3.4. b) Probabilities of the mass M at the velocity FHB stars in our sample (with apparent magnitudes best-fitting β of 1.25, including Leo I (solid line) and excluding Leo I V < 16) that are responsible for setting the minimum mass of − 1 (dashed line). the Galaxy have proper-motion errors of 5masyr− , whereas Draco and Pal 3 have much larger relative∼ errors, comparable to their proper motions themselves (see Table 2). Indeed, both values of the total mass. Thus, about 24% of the total mass of the Space Interferometry Mission (SIM: Unwin et al. 1997) and the Galaxy resides within r 50 kpc. This implies that the the Global Astrometry Interferometer for Astrophysics (GAIA: possibility of brown dwarfs as≤ the origin of the microlensing Lindegren & Perryman 1996) will be able to provide more ac- events toward the LMC may be excluded, because it requires a curate proper motions for such high-velocity objects, as well much smaller mass inside 50 kpc, 1.3 1011 M (Honma & as for numerous other distant tracers of the Galaxy’s mass, Kan-ya 1998). Our result is also in good∼ agreement× with the re- up to a precision of a few µas for targets with V 15. This 1 ≤ cent statistics of the microlensing events obtained from analysis corresponds to an error of <10 km s− in the tangential ve- of the 5.7-year baseline of photometry for 11.9 million stars in locity components for many∼ distant objects, i.e., comparable the LMC (Alcock et al. 2000), showing the absence of short- to the error of their (presently determined) radial velocities. duration lensing events by brown dwarfs. However, the most Furthermore, roughly half of our sample objects lack proper- recent work has suggested that perhaps one of the microlens- motion measurements altogether. To a great extent, the lack ing events is actually caused by a nearby low-mass star in the of proper-motion measurements (at least for southern sources) 910 T. Sakamoto et al.: The mass of the Milky Way will be removed with the completion of the recently re-started yet undetermined) mixture of binaries and high-gravity stars Southern Proper Motion survey of van Altena and colleagues, (see Preston & Sneden 2000). For some applications, such as as well as other efforts to substantially increase the numbers estimates of the mass of the Galaxy that rely on space mo- of stars with reasonably well-measured proper motions (e.g., tions of tracers (and in turn on reasonably precise distance UCAC1: Zacharias et al. 2000; UCAC2: Zacharias et al. 2001). estimates of individual objects), confident separation of bona- Further assembly of radial velocities for FHB stars, espe- fide members of the FHB population from possible “contami- cially those at large r (beyond distances where accurate ground- nants” is crucial3. In the past, this has required that one obtain based proper motions can be obtained), is also of great impor- either Strömgren photometry and/or spectrophotometry (e.g., tance for a number of reasons. First, as Fig. 3 demonstrates, Kinman et al. 1994), broad-band UBV photometry in combina- large Galactocentric regions are characterized by small es- tion with medium-resolution spectroscopy (e.g., Wilhelm et al. cape velocities. The current sample of FHB stars (because of 1999a), or reasonably high S/N, high-resolution spectroscopy their locations near the Sun) explore distances where the cor- (e.g., Preston & Sneden 2000). All such endeavors are rather responding escape velocities are in the range of 500 < Vesc < time intensive. However, Christlieb et al. (2002, priv. comm.) 1 ∼ ∼ 600 km s− . More remote FHB stars, with distances in the range have been exploring means by which adequate separation of 10 < r < 50 kpc, will offer a further constraint on the total mass FHB stars from higher-gravity A-type stars might be accom- ∼ ∼ 1 of the Galaxy by covering the range 400 < Vesc < 500 km s− . plished directly from objective-prism spectra, such as those in Secondly, the assembly of samples of more∼ distant∼ FHB stars the Hamburg/ESO stellar survey. Such methods, which look will enable exploration of the suggested change in velocity promising, would be most helpful in future investigations of anisotropy from the inner to the outer halo (e.g., Sommer- this sort. Wide-field stellar surveys, such as those presently Larsen et al. 1997), and better constrain its dependence on being carried out with the 6dF facility at the UK Schmidt Galactocentric distance. Telescope, are capable of providing large numbers of radial ve- In exploring the Bayesian approach for mass estimates of locities for FHB/A candidates, and are expected to contribute the Galaxy, we have adopted a specific form of the phase-space 5000–10000 suitable data over the course of the next few years. distribution function F (Eq. (8)) to facilitate comparison with previous studies. This procedure implicitly assumes that the Acknowledgements. We are grateful to B. Fuchs, R. B. Hanson, and I. velocity-anisotropy parameter, β, is constant everywhere in the Platais for assistance with the comparison of the Wilhelm et al. sam- Galactic volume. However, as noted by Sommer-Larsen et al. ple with the catalogs of STARNET, NPM1, and SPM 2.0, respec- (1997), there is an indication that the velocity anisotropy of tively. We also thank the members of the VERA team for several useful comments on this work. T.C.B. acknowledges partial support the halo may be mostly radial at R < 20 kpc and tangential from grants AST-00 98508 and AST-00 98549 awarded by the U.S. at R > 20 kpc. If so, many of distant∼ FHB stars, especially ∼ National Science Foundation. T.C.B also would like to acknowledge those at R > 20 kpc, play a crucial role in the determination the support and hospitality shown him during a sabbatical visit to the of the global distribution of velocity anisotropy. Searches for National Astronomical Observatory of Japan, funded in part by an in- a more realistic form of the phase-space distribution function, ternational scholar award from the Japanese Ministry of Education, combined with a more elaborate likelihood method, are both Culture, Sports, Science, and Technology, during which initial discus- worthy pursuits. Also, instead of exploring such a specific but sions of this work took place. realistic form of distribution function, a non-parametric method as proposed by Merritt & Tremblay (1993) will be more useful References if a much larger data set is available. Moreover, the implicit assumption behind the current Alcock, C., Allsman, R. A., Alves, D. R., et al. 2000, ApJ, 542, 281 method, that the sample stars have random distributions in lo- Alcock, C., Allsman, R. A., Alves, D. R., et al. 2001, Nature, 414, 617 cation and in their space motions, may not be well satisfied if Anguita, C., Loyola, P., & Pedreros, M. H. 2000, AJ, 120, 845 the halo is largely dominated by coherent structures such as Beers, T. C., Chiba, M., Yoshii, Y., et al. 2000, AJ, 119, 2866 tidal streams (e.g., Ibata et al. 2001). Alternative approaches to Binney, J. J., & Tremaine, S. 1987, Galactic Dynamics (Princeton obtaining mass limits using tidal streams (Johnston et al. 1999) Univ. Press, Princeton) Carney, B. W., Laird, J. B., & Latham, D. W. 1988, AJ, 96, 560 are worth considering in such a case. Carretta, E., Gratton, R. G., & Clemintini, G. 2000, MNRAS, 316, Fortunately, prospects are excellent for obtaining a rapid 721 increase in the observational database of FHB stars with the Carretta, E., Gratton, R. G., Clementini, G., & Pecci, F. F. 2000, ApJ, required data. There already exists a substantial body of ad- 533, 215 ditional spectroscopy for FHB/A stars observed during the Chen, L., Geffert, M., Wang, J. J., Reif, K., & Braun, J. M. 2000, course of the HK survey of Beers and colleagues and the A&AS, 145, 223 Hamburg/ESO Stellar survey (Christlieb et al. 2001), many of Chiba, M., & Beers, T. C. 2001, ApJ, 549, 325 which also have available proper motions, or will soon, from Christlieb, N., Wisotzky, L., Reimers, D., et al. 2001, A&A, 366, 898 completion of the SPM survey and/or other ground-based ef- Côté, P., Mateo, M., Sargent, W. L. W., & Olszewski, W. 2000, ApJ, forts. However, as was noted by Wilhelm et al. (1999a) (fore- 537, L91 shadowed by Norris & Hawkins 1991; Rodgers & Roberts Dauphole, B., & Colin, J. 1995, A&A, 300, 117 1993, and references therein; Kinman et al. 1994; Preston 3 For example, if 10% of our FHB sample is contaminated by blue et al. 1994), a substantial fraction (perhaps as high as 50%) metal-poor stars, we obtain a 2 3 1011 M decrease in our total of high-latitude A-type stars are not FHB, but rather some (as mass estimate, based on Monte Carlo∼ × experiments. T. Sakamoto et al.: The mass of the Milky Way 911

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